"can be elements of a continuous distribution, such as a solid body. Rearranging equation () by vector identities, multiplying both terms by ""one"", and grouping appropriately, gives the total angular momentum of the system of particles in terms of moment of inertia formula_5 and angular velocity formula_124, In the case of a single particle moving about the arbitrary origin, For the case of the center of mass fixed in space with respect to the origin, In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of"
"Momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction in three-dimensional space. If is an object's mass and is the velocity (also a vector), then the momentum is In SI units, it is measured in kilogram meters per second (kg⋅m/s). Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial"
"acceleration is equal to the sum of the forces on the mass. Newton's law of gravity says that the gravitational force felt on mass by a single mass is given by where is the gravitational constant and is the magnitude of the distance between and (metric induced by the norm). Summing over all masses yields the -body equations of motion: where is the ""self-potential"" energy Defining the momentum to be , Hamilton's equations of motion for the -body problem become where the Hamiltonian function is and is the kinetic energy Hamilton's equations show that the -body problem is a system"
"concept can also be thought of as a classical physics analogue of the quantum mechanical concept of quasiparticles. It is, however, not to be confused with relativistic mass increase. It is often erroneously stated that the added mass is determined by the momentum of the fluid. That this is not so becomes clear when considering the case of the fluid in a large box, where the fluid momentum is exactly zero at every moment of time. The added mass is actually determined by the quasi-momentum: the added mass times the body acceleration is equal to the time derivative of the"
"mass. This was clear at the beginning of the 20th century, before the advent of the theory of relativity. Mass, ""m"", denoted something like an amount of substance or quantity of matter. At the same time, mass was the quantitative measure of inertia of a body. The mass of a body determines the momentum, formula_1, of the body at given velocity, formula_2; it is a proportionality factor in the formula: The factor ""m"" is referred to as inertial mass. But mass, as related to the ""inertia"" of a body, can also be defined by the formula: where ""F"" is the"
"pendulum mass around the pivot, where formula_40 is the angular velocity of the mass about the pivot point. This angular momentum is given by using a similar derivation to the previous equation. Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield This shows that the quantity formula_37 is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values formula_1 for all of the elements of mass in the body. A"
"events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass ""m"" moves at a velocity v at an initial time ""t"". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time ""t"" + d""t"", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d""m""dv is the product of two small values, it can be ignored, meaning during d""t"""
"describe the same properties of physical bodies at the qualitative and quantitative level respectively, by other words, the mass quantitatively describes the inertia. According to Newton's second law of motion, if a body of fixed mass ""m"" is subjected to a single force ""F"", its acceleration ""a"" is given by ""F""/""m"". A body's mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass ""m"" is placed at a distance ""r"" (center of mass to center of mass) from a second body of mass ""m"", each body is subject"
"(say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations. Additionally, mass relates a body's momentum p to its linear velocity v: and the body's kinetic energy ""K"" to its velocity: The primary difficulty with Mach's definition of mass is that it fails to take into account the potential energy (or binding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass. This is most vividly demonstrated by comparing the mass of the proton in the"
"""a"", ""a"") can be calculated from Newton's Law of Gravitation. Each body attracts each other body, the total acceleration being the sum of all these attractions. Next one chooses a small time-step Δ""t"" and applies Newton's Second Law of Motion. The acceleration multiplied with Δ""t"" gives a correction to the velocity. The velocity multiplied with Δ""t"" gives a correction to the position. This procedure is repeated for all other bodies. The result is a new value for position and velocity for all bodies. Then, using these new values one starts over the whole calculation for the next time-step Δ""t"". Repeating"
"of inertia about an axis of a body is calculated by summing formula_1 for every particle in the body, where formula_2 is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.) Consider the kinetic energy of an assembly of formula_74 masses formula_75 that lie at the distances"
"where u is the velocity of the ejected mass. Thus during d""t"" the momentum of the system varies for This is the same dp as found in the mass accretion case above, meaning that the same conclusion holds. By the definition of acceleration, a = dv/d""t"", so the variable-mass system motion equation can be written as In bodies that are not treated as particles a must be replaced by a, the acceleration of the center of mass of the system, meaning Often the force due to thrust is defined as formula_12 so that This form shows that a body can"
"mass ""m"" varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is where u is the velocity of the escaping or incoming mass relative to the body. From this equation one can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket equation. Under some conventions, the quantity u d""m""/d""t"" on the left-hand side, which represents the advection of momentum, is defined as a force (the force exerted on the body by the"
"force, ""m"" is the inertial mass, and ""a"" is the acceleration. By this formula, the greater its mass, the less a body accelerates under a given force. The masses defined by the above formulas are equal because the latter formula is a consequence of the former, if mass does not depend on time and velocity. Thus, ""mass is the quantitative or numerical measure of a body’s inertia, that is of its resistance to being accelerated"". This meaning of a ""body's inertia"" therefore is altered from the popular meaning as ""a tendency to maintain momentum"" to a description of the measure"
"the mass of the body as well as its speed. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. In formula form: where formula_2 is the mass and formula_3 is the speed (or the velocity) of the body. In SI units, mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as When a person"
"the body and formula_2 is the distance from the pivot point formula_51 to the center of mass formula_60. Moment of inertia of a body is often defined in terms of its ""radius of gyration"", which is the radius of a ring of equal mass around the center of mass of a body that has the same moment of inertia. The radius of gyration formula_19 is calculated from the body's moment of inertia formula_55 and mass formula_3 as the length A simple pendulum that has the same natural frequency as a compound pendulum defines the length formula_5 from the pivot to"
"acceleration. Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer. In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law): where F is the net force acting on the body, ""m"" is the mass of the body, and a is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large. The velocity of a particle moving on a"
"respective masses and , and velocities and , the total momentum is The momenta of more than two particles can be added more generally with the following: A system of particles has a center of mass, a point determined by the weighted sum of their positions: If all the particles are moving, the center of mass will generally be moving as well (unless the system is in pure rotation around it). If the center of mass is moving at velocity , the momentum is: This is known as Euler's first law. If the net force applied to a particle is"
"human scale. By comparison, running with a five-pound object (10×Planck mass) at an average running speed (10×speed of light in a vacuum) would give the object Planck momentum. A 70 kg human moving at an average walking speed of would have a momentum of about 15 formula_1. A baseball, which has mass formula_9 0.145 kg, travelling at would have a Planck momentum. Planck momentum Planck momentum is the unit of momentum in the system of natural units known as Planck units. It has no commonly used symbol of its own, but can be denoted by formula_1, where formula_2 is the"
"velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s). In cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second (g⋅cm/s). Being a vector, momentum has magnitude and direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground. The momentum of a system of particles is the vector sum of their momenta. If two particles have"
"Mass Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does"
"Another three base units (for temperature, amount of substance and luminous intensity) were added later. The early metric systems defined a unit of weight as a base unit, while the SI defines an analogous unit of mass. In everyday use, these are mostly interchangeable, but in scientific contexts the difference matters. Mass, strictly the inertial mass, represents a quantity of matter. It relates the acceleration of a body to the applied force via Newton's law, : force equals mass times acceleration. A force of 1 N (newton) applied to a mass of 1 kg will accelerate it at 1 m/s."
"formula_76 from the pivot point formula_51, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses, This shows that the moment of inertia of the body is the sum of each of the formula_1 terms, that is Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia. The moment of inertia of a continuous body rotating about a specified axis is calculated"
"displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the force applied to the body; rearranging this derivative statement leads to the famous equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called"
"is acted upon by a force, as described. Momentum is a quantity which is used for measuring the motion of an object. An object's momentum is directly related to the object's mass and velocity, and the total momentum of all objects in an isolated system (one not affected by external forces) does not change with time, as described by the law of conservation of momentum. As there is no absolute frame of reference, ""absolute motion"" cannot be determined. Thus, everything in the universe can be considered to be moving. Motion applies to objects, bodies, and matter particles, to radiation, radiation"
"to the angular acceleration formula_9 around a principal axis, that is For a simple pendulum, this definition yields a formula for the moment of inertia formula_4 in terms of the mass formula_3 of the pendulum and its distance formula_2 from the pivot point as, Thus, moment of inertia depends on both the mass formula_3 of a body and its geometry, or shape, as defined by the distance formula_2 to the axis of rotation. This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses formula_17 each multiplied by"
"the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system. Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved. In a system of two particles, if formula_14 is the momentum of object 1 and formula_15 the momentum of object 2, then Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define"
"pressed together. Friction always acts parallel to the surface in contact and opposite the direction of motion. The friction force can be calculated using the equation. A force formula_3, dependent on time formula_4, acting on a body of assumed constant mass formula_5 for a time interval formula_6 generates a change in the body’s momentum formula_7, where formula_8 is the resulting change in velocity. The change in momentum, termed an impulse and denoted by formula_9 is thus computed as For fixed impulse formula_10, the equation suggests that formula_11, that is, a smaller time interval must be compensated by a stronger reaction"
"Moment of inertia The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate. It is an extensive (additive) property: for a point mass the moment of inertia is just the mass times the square of the perpendicular distance to the rotation"
"body by mass and volume (mass divided by volume gives the mass density of the body). In a gravitational (or other body force) field, and in a vacuum, the body experiences a transfer of momentum giving its weight. If the vacuum is substituted by a medium (water, air), then the body loses the weight of the displaced medium's volume (otherwise that volume of medium replaced into the remaining medium would not remain force-free). Reconsider the body and the body's volume made up by medium: Subtracting the respective vacuum-weights (body minus medium) gives the resultant, effective, net-weight of the body in"
"- both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by formula_1, where formula_2 is the distance of the point from the axis, and formula_3 is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in question. For"
"components. The total momentum formula_20 of the system, a vector quantity, can also be computed by adding together the momenta of all its components. Given the total energy ""E"" and the length (magnitude) ""p"" of the total momentum vector formula_20, the invariant mass is given by: In a mathematical system where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following: Where, again, the particle momenta formula_20 are first summed as vectors, and then the square of their resulting total magnitude (Euclidean norm) is used. This results in"
"separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is : it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes . The momentum is equal to the energy multiplied by the velocity divided by ""c"". As such, the Newtonian mass of an object,"